Thread: going in circles

Hi folks,

I am trying to solve the following question:

Find the equations of the two circles which pass through the point (2,0) and have both the y-axis and the line y - 1 = 0 as tangents.

This seems like an easy question because you can do it by scale drawing. The first (small) circle can't really go anywhere else - see attachment! But assuming you had to do it by calculation.

To find the equation of a circle we need 3 points on the circumference. We have one supplied directly (2,0) then we have the general equation:



plus the two tangents:

x = 0

y = 1

substituting (2,0) gives:

4 + 4f + c = 0

and substituting x = 0 into the general equation gives:



a quadtratic in y. Since the tangent touches the circle we expect a repeated root i.e.






then substitute y = 1 into the general equation:





again setting gives:




Equations (4) and (5) are correct. What I mean is that when you substitue the correct answers in, they give consistent results but how do you solve them? It seems clear that I am on the wrong track. I then realised that the equation y = -x + 1 must be a diameter to both circles so I tried substituting this into the general equation (1) to solve for the intersection points. It yielded a result for x with many terms in f and g that I couldn't reduce to anything like the correct answer.

Any help here would be appreciated!
best regards

Hi

AS you said the line y=-x+1 is a diameter of the circles
The centers are on this line
Let C (xC,yC) one center of the possible circles
yC = -xC + 1

The distance between C and the point M(2,0) is such that CM² = (xC-2)²+yC² = 2xC² - 6 xC + 5
This distance is the same as the one between C and the y-axis, which is |xC|
Therefore : 2xC² - 6 xC + 5 = xC²
xC² - 6 xC + 5 = 0
(xC - 1)(xC - 5) = 0
xC = 1 or 5

Originally Posted by s_ingram

Hi folks,

I am trying to solve the following question:

Find the equations of the two circles which pass through the point 2,0) and have both the y-axis and the line y-1=0 as tangents.

...

1. I used this equation of a circle:

with the center at C(a, b) and the radius r.

2. You already know 3 points of the circle: (2, 0), (0, 1-r), (a, 1)

3. Substitute the coordinates of these points into the equation of the circle. You'll get a system of equations. Solve this system for a, b and r.



4. After some transformations and calculations you'll get:

Circle #1:
Circle #2:

Hello, s_ingram!

Find the equations of the two circles which pass through the point (2,0)
and have both the -axis and the line as tangents.

Code:

        |
        |         R
      - + - P - * o * - - - y=1
    ----+---o-----:-----*----------
        | * 2     :       *
        |*        :        *
        |         :
        *         :         *
      Q o - - - - o C       *
        *       (x,y)       *
        |
        |*                 *
        | *               *
        |   *           *
        |       * * *
        |

The center of the circle is

It passes through the point
It is tangent to the -axis at
It is tangent to at

The radius is: .


We have: .

Since , then: .

Since , we have: .

. . which becomes: .

Then: .

. . which simplifies to: .

. . which factors: .

. . and has roots: .

. . . . . . . . Then: .


One circle has center and

The other has center and

Excellent! Thanks a lot guys.